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Perfect square and Positive Integer | TOMATO B.Stat Objective 115

Try this TOMATO problem from I.S.I. B.Stat Entrance Objective Problem based on Perfect square and Positive Integer. You may use sequential hints.

Try this problem from I.S.I. B.Stat Entrance Objective Problem based on Perfect square and Positive Integer.

Perfect Square & Positive Integers ( B.Stat Objective Question )


If n is a positive integer such that 8n+1 is a perfect square, then

  • n must be odd
  • 2n cannot be a perfect square
  • n must be a prime number
  • none of these

Key Concepts


Perfect square

Positive Integer

Primes

Check the Answer


But try the problem first…

Answer: 2n cannot be a perfect square.

Source
Suggested Reading

B.Stat Objective Problem 115

Challenges and Thrills of Pre-College Mathematics by University Press

Try with Hints


First hint

Let \((8n+1)=k^{2}\) be a perfect square so k is found to be odd here as 8n+1 is odd.

Second Hint

\(\Rightarrow 8n=k^{2}-1\)

\(\Rightarrow 8n=(k-1)(k+1)\)

\(\Rightarrow 2n=\frac{(k-1)(k+1)}{4}\)

Final Step

\(\Rightarrow (\frac{k-1}{2})(\frac{k+1}{2})\)

here (k-1) and (k+1) are consecutive even numbers then \((\frac{k-1}{2})(\frac{k+1}{2})\) are consecutive even numbers

\(2k \times (2k+2)= 2^2{k(k+1)}\) is not a perfect square as product of two consecutive numbers proved here as not a perfect square

So, 2n is not perfect square.

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